322 ANALYTICAL MECHANICS (XXII) is the general solution of equation (2). Now let 6 = 0 when t = 0, then c2 = Ci. Therefore 6 = Atf~ at(eVa*"bn <3~ v/a" ~'''"'), (XXI) where Ai = aci. There are three special cases which must be discussed separately. Case I. Let a2 = 62, then 0 = 0 for all values of the time. Therefore this is a case of no motion. Case II. Let a2 > 62, then Va2 b'2 is real. Denoting this radical by c we have The character of the motion is brought out by the graph of equation (XXII), Fig. 145. The graph is easily obtained by drawing the dotted curves, which are _ plotted by considering the ° terms of the right-hand member of equation (XXII) separately, and then adding them geometrically. It is evident from the curve that the value of. B starts at zero, FuL I4r>- increases to a maximum, and then diminishes to zero asymptotically. In this case the motion is said to be aperiodic or dead-beat. Case III. Let_o^<62 then Va» V - i = i and Vb* - a2 = Then this substitution in equation (XXI) we obtain fl'-2i sin /- is imaginary. Let ~ Making (XXIII) Sec Appendix Avu.